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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Try a solution of the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_13.html ./knowl/eq7_12_1.html">
\begin{equation*}
{\bf x}^{(2)}=(t \vec{\xi}+\vec{\eta}) e^{r_1 t}=t \vec{\xi}+\vec{\eta},
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(\vec{\xi}, \vec{\eta}\)</span> are to be determined. Substituting it into (<a href="" class="xref" data-knowl="./knowl/eq7_13.html" title="Equation 6.4.4">(6.4.4)</a>):</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_13.html ./knowl/eq7_12_1.html">
\begin{equation*}
\vec{\xi}={\bf A} (t\vec{\xi}+\vec{\eta})=t {\bf A} \, \vec{\xi}+{\bf A}\, \vec{\eta}.
\end{equation*}
</div>
<p class="continuation">Compare both sides:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_13.html ./knowl/eq7_12_1.html">
\begin{equation}
{\bf A} \, \vec{\xi}={\bf 0},\quad
{\bf A} \, \vec{\eta}=\vec{\xi}.\tag{6.4.8}
\end{equation}
</div>
<p class="continuation"><span class="process-math">\((\ref{eq7_15})_1\)</span> is the same as (<a href="" class="xref" data-knowl="./knowl/eq7_12_1.html" title="Equation 6.4.6">(6.4.6)</a>), thus</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_13.html ./knowl/eq7_12_1.html">
\begin{equation*}
\vec{\xi}=\vec{\xi}^{(1)}=\left(
\begin{array}{c}
1\\
2
\end{array}
\right).
\end{equation*}
</div>
<p class="continuation">From <span class="process-math">\((\ref{eq7_15})_2\text{,}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_13.html ./knowl/eq7_12_1.html">
\begin{equation*}
\left(
\begin{array}{cc}
4 &amp; -2\\
8 &amp; -4
\end{array}
\right) \left(
\begin{array}{c}
\eta_1\\
\eta_2
\end{array}
\right)=\left(
\begin{array}{c}
1\\
2
\end{array}
\right) \to \eta_2=2 \eta_1-1/2.
\end{equation*}
</div>
<p class="continuation">Choose <span class="process-math">\(\eta_1=0\text{,}\)</span> we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq7_13.html ./knowl/eq7_12_1.html">
\begin{equation*}
\vec{\eta}=\left(
\begin{array}{c}
0\\
-1/2
\end{array}
\right).
\end{equation*}
</div>
<span class="incontext"><a href="sec6_4.html#p-275" class="internal">in-context</a></span>
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